This one seems to be anonymous from the little I know about it. Parodies were popular then as now. 'Little Billee' is a parody of the French 'La Courte Paille' or 'Le Petit Navire' (Thackeray apparently used to sing his poem to that tune) rather than 'The Ship in Distress', but of course the latter is based on the French song, which in turn was probably based on a C17 Scandinavian one.

There were other parodies of 'Lord Lovel', and also of 'George (Giles) Collins', 'Green Willow' ('All Around My Hat') and so. Steve Gardham is interested in the genre and may well turn up here in due course; he will be able to add a lot more.

Meanwhile, what tune did Norman use? Logically it would be a 'Lord Lovel' variant, but that would depend on where he got it. I don't know of any survivals in tradition of the 'Muggins' parody, so he (or whoever he learned it from) may have put a convenient melody to a broadside text.